In order to derive realistic predictions of the prompt atmospheric neutrino flux at the
detector level we should have theoretical control of the description of
several ingredients (see Fig. REF ): the incident cosmic
flux, the charm production, its hadronization, the decay of the
heavy hadrons, the propagation of the associated particles
through the atmosphere and the neutrino interaction (see e.g.
Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}).
As demonstrated in our previous study [10]}, to address the production of
high-energy neutrinos (\(E_{\nu } >\)  10\(^5\)  GeV), it is fundamental to
precisely describe the charmed meson production at very high energies
and large forward rapidities. This aspect motivated the development
of new and/or more precise approaches to describe the perturbative and
nonperturbative regimes of the Quantum Chromodynamics (QCD) needed to
describe the charmed meson production in a kinematical range beyond
that reached in hadronic collisions at the LHC. For this new kinematical
range, some topics are theme of intense debate: (a) the presence (or
not) of intrinsic heavy quarks in the hadronic wave function
[11]}, [12]}, [13]}, characterized by a large value of
the longitudinal momentum fraction of beam nucleon momentum;
(b) the validity of the collinear factorization at high energies
[14]}, [15]}, [16]}, [17]}, since it disregards
the transverse momentum of the incident particles; and (c) the presence
(or not) of nonlinear (saturation) effects on the description of the QCD
dynamics at high energies [18]}, which are expected to contribute
at high energies due to the high partonic density predicted by linear
DGLAP or BFKL evolution equations;
(d) the impact of subleading fragmentation of light partons
on heavy meson production at high energies and very forward rapidities
and its consequences for prompt neutrino flux [19]}, [9]}.
Such questions naturally arise due to the fact that
in the calculation of the prompt neutrino flux at high energies, the
main contribution for the charm production cross section comes from
partons with very small (large) values of \(x\)  in the hadrons that
constitute the atmosphere (incident cosmic ray flux). Recently, two of
us have presented in Ref. [21]} a comprehensive study
of the charm production at large rapidities considering the collinear,
hybrid and \(k_T\) -factorization approaches taking into account
the presence of an intrinsic charm in the proton wave function with
parton distributions that are solutions of linear and nonlinear
evolution equations. One of the goals of this paper is to extend the
analysis performed in Ref. [21]} and derive
associated prompt neutrino fluxes at high energies. In particular, we
shall estimate the impact of the intrinsic charm – initiated subprocess
and/or saturation effects on the predictions for the prompt neutrino flux.
Another more ambitious goal is to verify whether the recent IceCube data
for the prompt \(\nu _{\mu }\)  flux allow to derive an upper bound for the probability
of finding a charm quark-antiquark pair in the proton wave function,
which is one of the main uncertainties in the modelling of
the intrinsic charm. A similar goal was also present in the
analyses performed in Refs. [4]}, [6]}. However, our study
differs from these previous analyses in several aspects.
Our predictions for the \(x_F\)  distributions will be derived using a
framework that sucessfully describes the LHC data, with the main input
being the parton distribution functions which were derived using the
world data.
In these previous studies, the \(x_F\)  distribution was fitted using the
old data for the \(D\)  and \(\Lambda _c\)  production, with the normalization
being a parameter free.
Moreover, the energy dependence of the intrinsic charm contribution was
assumed to follow the inelastic cross section, which is dicted by soft
processes. In contrast, in our approach, such contribution is
calculated perturbatively, which implies a steeper energy dependence.
Finally, our predictions are calculated using a unified approach for
the \(gg \rightarrow c \bar{c}\)  and \(gc \rightarrow gc\)  mechanisms,
which we believe to be more realistic to treat the charm production
at central and forward rapidities.
